Evolution divergent phase

The classical problem is no less daunting The initial state is defined in terms of the 3N coordinates and 3N momenta of the N particles, which together define a 6iV-dimensional phase space. Each point in this phase space defines a specific microstate. Evolution of the system is described by the trajectory of the starting point through phase space. Not only is it beyond the capabilities of the most powerful computer to keep track of these fluctuations, but should only one of the starting parameters be defined incorrectly, the model fatally diverges very rapidly from the actual situation. 

The present study should be seen as a step in the evolution of the colloidal morphology of phase inversion membranes, which conceptually began with dense polymer films and diverged into the two principal branches skinned and skinless membranes (Figure 1). 

We will discuss this state in relation to the recent approaches of the anomalous diffusion theory [31]. It is well known [226-230] that by virtue of the divergent form of Poisson brackets (95) the evolution of the distribution function pip,q t) can be regarded as the flow of a fluid in phase space. Thus the Liouville equation (93) is analogous to the continuity equation for a fluid... 

In the phase space, the trajectory followed by the system never passes again through the same point, but remains confined to a finite portion of this space (fig. 4.10) the system evolves towards a strange attractor (Ruelle, 1989). The unpredictability of the time evolution in the chaotic regime is associated with the sensitivity to initial conditions two points, initially close to each other on the strange attractor, will diverge exponentially in the course of time. 

Consider a set of points f) in phase space with evolution associated to a differential equation z = /(z) described by the flow map f(S(0)) = > t). Liouville s theorem [ 16] states that the volume of such a set is invariant with respect to t if the divergence of / vanishes, i.e. 

A necessary condition for the inequality (2.76) to be realizable is p cos(Tr/n) > 1 which means, e.g., p > 8 for n = 3 and p > 4 for n = 4. If this condition is fulfilled, a sufficiently large value of Q, i.e., a sufficiently large substrate concentration S, cf. (2.62), will then cause at least two of the modes of (2.73) to leave the steady state in a spiral motion. The question of what eventually happens to these instable modes cannot be answered within the framework of our differential analysis of the steady state. On the other hand, we do not expect that these modes really diverge to infinite values of since the feedback coupling of our model will prevent such an explosion. As a matter of fact, one can show, (cf. TYSON (1975)), that the instable modes approach a so-called limit cycle, i.e., a stable, undamped periodic oscillation around the steady state. The instable spiral motion, which we have found in our differential analysis, thus represents the initial phase of the evolution towards the limit cycle. 

Because microbial rhodopsins are present in aU three domains of hfe, progenitors of these proteins may have existed in the early phases of evolution, before the divergence of archaea, eubacteria, and eukaryotes. 

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